A centrifuge can be used to separate cells from a liquid. To increase the rate at which objects can be separated from solution, it is useful to make the centrifuge's speed as large as possible. If you want to design a centrifuge of diameter 1.4 m to have a force of 128,000 times the force of the Earth's gravity, what is the speed of the outer edge of the centrifuge? Such a device is called an ultracentrifuge. Centrifuges of this type are currently being used by certain countries (including Iran) to produce what is called enriched uranium for use in nuclear power plants or weapons. Naturally occurring uranium contains two isotopes (which have slightly different masses), only one of which is useful in these applications (see a later chapter). Ultracentrifuges like the one considered in this problem are used to process the gas UF6 and thereby separate the two uranium isotopes.

To find the speed of the outer edge of the centrifuge, we can use the formula for centrifugal force:

F = m * r * ω²

Where:
F is the centrifugal force
m is the mass of an object
r is the radius of rotation
ω is the angular velocity

In this case, the desired centrifugal force is 128,000 times the force of Earth's gravity. We can calculate this force using the following equation:
F = 128,000 * g

Where:
g is the acceleration due to gravity (9.8 m/s²)

Now, let's calculate the radius of rotation:
r = diameter / 2
r = 1.4 m / 2

Next, we need to calculate the mass of an object. Considering a small object, we can use the equation:
m = ρ * V

Where:
ρ is the density of the object
V is the volume of the object

For simplicity, let's assume the density of the object is the same as the liquid. We can also assume the object has a spherical shape, so the volume can be calculated using the equation for the volume of a sphere:
V = (4/3) * π * r³

Now we can determine the angular velocity ω:
ω = √(F / (m * r))

Let's put all the values into the equations and calculate step by step:

1. Calculate F:
F = 128,000 * 9.8 = 1,254,400 N

2. Calculate the radius:
r = 1.4 / 2 = 0.7 m

3. Calculate the volume:
V = (4/3) * π * (0.7)³ = 1.437 m³

4. Calculate the mass: (assuming ρ = density of the liquid)
m = ρ * V

5. Calculate the angular velocity:
ω = √(F / (m * r))

Finally, the calculated angular velocity will give us the speed of the outer edge of the centrifuge.

To determine the speed of the outer edge of the centrifuge, we need to calculate the angular velocity (ω) first. The equation for centrifugal force is given by:

Fc = m * ω^2 * r

Where Fc is the centrifugal force, m is the mass being separated, ω is the angular velocity, and r is the radius. In this case, we want the centrifugal force to be 128,000 times the force of the Earth's gravity, which we can represent as:

Fc = 128,000 * g

Where g is the acceleration due to gravity (approximately 9.8 m/s^2). Since the centrifuge's radius is given (1.4 m), we can rearrange the equation and solve for ω:

128,000 * g = m * ω^2 * r
128,000 * 9.8 = m * ω^2 * 1.4

We also know the formula for calculating angular velocity:

ω = v / r

Where v is the linear velocity. Substituting the value of ω into the equation, we have:

128,000 * 9.8 = m * (v / r)^2 * 1.4

Simplifying, we get:

v^2 = (128,000 * 9.8 * r) / m
v = sqrt((128,000 * 9.8 * r) / m)

Since we are not provided with the mass (m) being separated or any other relevant information, we cannot determine the exact speed. To calculate the speed of the outer edge of the centrifuge, you will need to know the mass being separated and substitute it into the equation above.